Question

A long wire is uniformly charged with linear charge density $\lambda$ and have a semi-circular section of radius $r$. The magnitude of electric field at the centre of circular arc is

• A. $\frac{\lambda}{4\pi\epsilon_0r}$

Answer 1

Answer: (A)

$\frac{\lambda}{4\pi\epsilon_0r}$

Answer 2

The U-shaped wire consists of two semi-infinite straight segments and a semi-circular arc. The electric field at the center of the arc is the vector sum of fields from these three parts.

1. Electric field from the semi-circular arc: For a uniformly charged semi-circular arc of radius $r$ and linear charge density $\lambda$, the electric field at its center is $E_{arc} = \frac{\lambda}{2\pi\epsilon_0 r}$ directed perpendicular to the diameter, away from the arc. In this setup, it's directed upwards ($\hat{j}$).
2. Electric field from the two straight semi-infinite wires: Each semi-infinite wire produces an electric field at the center of the arc. For a semi-infinite wire, the field components perpendicular and parallel to the wire (at a point perpendicular to its end) each have a magnitude of $\frac{\lambda}{4\pi\epsilon_0 r}$. Due to symmetry, the horizontal components from the two straight wires cancel out. The vertical components from both straight wires add up and are directed downwards (towards the ends of the wires). The total vertical field from the two straight wires is $E_{straight} = 2 \times \frac{\lambda}{4\pi\epsilon_0 r} = \frac{\lambda}{2\pi\epsilon_0 r}$, directed downwards ($-\hat{j}$).
3. Total Electric Field: The total electric field is the vector sum of the field from the arc and the straight wires: $\vec{E}_{total} = \frac{\lambda}{2\pi\epsilon_0 r}\hat{j} - \frac{\lambda}{2\pi\epsilon_0 r}\hat{j} = \vec{0}$.

Since the calculated total electric field is zero and zero is not an option, there might be an error in the question or options. However, the value $\frac{\lambda}{4\pi\epsilon_0 r}$ is the magnitude of a single component of the electric field due to a semi-infinite wire. If the question implies a simplified scenario or asks for a specific part's contribution, this option could be considered. In the context of competitive exams, if the exact answer is not available, sometimes a related fundamental value is the intended answer due to question design flaws.